Multi-scale Heat Conduction Phonon Dispersion and Scattering Nov. 29th , 2011 Multi-scale Heat Conduction Phonon Dispersion and Scattering Hong goo, Kim 1st year of M.S. course

Contents Introduction Phonon Dispersion Phonon Scattering 1-D Diatomic Chain Phonon Branch Real Crystals Phonon Scattering Phonon-Phonon Process Anharmonic Effects Phonon-Defect Scattering Phonon-Electron Scattering Phonon-Photon-Electron Scattering Phonon-Photon Scattering

Phonon Concept Bose-Einstein Distribution Introduction Phonon Concept Quantized energy of lattice vibration Phonon is a boson with energy of ħω with to respect to the vibrational mode with frequency of ω Bose-Einstein Distribution Indistinguishable, unlimited # of particle per quantum state Specific Heat and Thermal Conductivity

Harmonic Wave Phase Velocity Group Velocity Introduction Harmonic Wave Phase Velocity Group Velocity For superposition of two waves with k1 ≈ k2 , ω1 ≈ ω2 vp = ω/k vg = Δω/Δk Modulation envelope Group velocity is the speed of energy propagation

1-D Diatomic Chain Assumptions Motion of the Atoms: F = ma = kx Phonon Dispersion 1-D Diatomic Chain Assumptions Displacement is sufficiently small → Linearity of atomic forces Only the nearest neighbor atoms interact each other m1 m2 C v2n-1 v2n+1 v2n+3 v2n+5 u2n-2 u2n u2n+2 u2n+4 Motion of the Atoms: F = ma = kx

1-D Diatomic Chain Harmonic Wave Solution: A exp( i(kx−ωt) ) Phonon Dispersion 1-D Diatomic Chain Harmonic Wave Solution: A exp( i(kx−ωt) ) a v2n−1 v2n+1 na x (n+0.5)a (n+1)a (n+2.5)a (n+3)a (n+3.5)a u2n u2n+2 (n−0.5)a Substitute

1-D Diatomic Chain Dispersion Relation Unknown : A1 and A2 Phonon Dispersion 1-D Diatomic Chain Dispersion Relation Unknown : A1 and A2 For nontrivial solution of A1 and A2 , determinant should be zero

1-D Diatomic Chain Dispersion Relation Phonon Dispersion ω Two branches are formed because of the difference between m1 and m2 Acoustic branch Optical branch Periodicity 2π/a of the reciprocal lattice space sin2(ka/2)={1−cos(ka)}/2 - Only the 1st Brillouin zone is needed k

Dispersion Relation Physical Meaning of Dispersion Relation Phonon Dispersion Dispersion Relation Physical Meaning of Dispersion Relation Relation between frequency(ω) and wavevector(k) In the presence of dispersion, phase velocity and group velocity is distinguished Characteristic of a material If dispersion relation is known, specific heat can be calculated (vg = dω/dk is known) Relation between energy(ħω) and momentum(ħk)

Phonon Branch Long wavelength limit : k → 0 Phonon Dispersion Phonon Branch Long wavelength limit : k → 0 Medium can be treated as a continuum Out-of-phase In-phase

Phonon Branch Long wavelength limit : k → 0 Acoustic branch : In-phase Phonon Dispersion Phonon Branch Long wavelength limit : k → 0 Acoustic branch : In-phase No change in relative motions between neighboring atoms Optical branch : Out-of-phase Restoring force acts within unit cells → high energy If atoms have different charges, oscillating electric dipole is produced Long wavelength가 대표하는 것은?

Phonon Branch Optical Branch Acoustic Branch Vibration within a cell Phonon Dispersion Phonon Branch Optical Branch ω k Optical Acoustic vg = dω/dk = 0 Vibration within a cell k → 0, vg = 0 ; standing wave, out-of-phase Interacts with EM waves vg > 0 From radiation theory, oscillating dipole scatters radiation Acoustic Branch Vibration of center of mass of a cell k → 0, vg > 0 ; running wave, in-phase, acoustic wave

Phonon Branch Number of Branches : q-atom unit cell Acoustic Optical Phonon Dispersion Phonon Branch Number of Branches : q-atom unit cell Acoustic Optical Longitudinal 1 q − 1 Transverse 2 2(q − 1) Longitudinal: atoms vibrate in the direction of wave propagation Transverse: atoms vibrate perpendicular to wave propagation Each cell has one LA branch and two TA branches For each additional atom, one LO branch and two TO branches are added Symmetry leads to degeneracy of transverse modes

Real Crystals Silicon Silicon Carbide Phonon Dispersion Real Crystals Silicon Silicon Carbide Si monatomic diamond-like structure LA meets LO (m1 = m2) TA, TO : degenerate Si & C diatomic structure Frequency gap exists D. W. Feldman et al.(1968) TO LO LA TA B. N. Brockhouse et al.(1959) TA LA LO TO Frequency gap

Real Crystals Silicon Phonon Dispersion TO LO LO LA TA Brillouin zone for silicon B. N. Brockhouse (1959) TA LA LO TO LO R. Tubino et al.(1971)

Real Crystals Optical Phonon Acoustic Phonon Phonon Dispersion Real Crystals Optical Phonon TA LA LO TO vg is small: slow propagation of phonons → less contribution on heat conduction Interaction with acoustic phonons at high temperature → reduction of thermal conductivity Significant contribution on heat capacity at high temperature With BE distribution, optical phonons (high frequency) get excited at higher temperature Acoustic Phonon Transverse acoustic (TA) Dominant mode at low temperature because low frequency modes are numerous Longitudinal acoustic (LA) More important at higher temperatures because upper limit of ω is higher than TA

Real Crystals Zeolite (Alx Siy Oz) Phonon Dispersion Real Crystals Zeolite (Alx Siy Oz) Nano-porous crystalline alumino-silicates Applications Sorption based heat exchanger: cooling of micro-electric devices Catalyst, molecular sieves for chemical separations Dielectric material MFI zeolite film 288 atoms per unit cell 864(=288×3) dispersion branches (polarization) Summation over all polarizations and wavevectors

Real Crystals: Measurement Phonon Dispersion Real Crystals: Measurement Neutron-Phonon Scattering Neutron beam is incident on the target material Emergent angle and energies of scattered neutron is measured Energy lost by neutron = absorption of phonon Conservation of crystal momentum Phonon dispersion can be derived EM(Photon-Phonon) Scattering Same conservation laws for neutron scattering holds X-ray scattering Visible: Raman(optical phonon), Brillouin(acoustic) scattering Very small frequency shift

Interaction of Phonons Phonon Scattering Interaction of Phonons Phonon Scattering Phonon is a convenient concept in describing thermal transport by lattice vibrational waves Phonons are treated as particles (wave → particle) Describes interaction of phonon with phonon/electron/defects and boundaries Anharmonic effect of phonon scattering governs the thermal transport properties of dielectric and semiconductor materials Phonon wave function can be localized by the uncertainty principle

Phonon-Phonon Three-Phonon Process Energy Conservation Phonon Scattering Phonon-Phonon Three-Phonon Process Dominant phonon-phonon scattering in terms of scattering probability 3rd order anharmonic term of interatomic potential Energy Conservation Simply a name for ħ times phonon wavevector Similarity with physical momentum in terms of expression and scattering behavior Crystal momentum is only conserved within the 1st Brillouin zone Crystal Momentum ħk Crystal Momentum Conservation

Phonon-Phonon Crystal Momentum Conservation N Process U Process k1 k1 Phonon Scattering Phonon-Phonon Crystal Momentum Conservation N Process U Process ky ky 1st Brillouin zone k1 k1 k2 k2 kx kx k3 G k3 k1 + k2

Phonon-Phonon Case 1 : N Process only Phonon Scattering Phonon-Phonon Case 1 : N Process only Net phonon momentum is conserved (G = 0) Nonzero phonon flux exists even without temperature gradient Equilibrium cannot be reached by N Processes only At Equilibrium, phonon momentum distribution is symmetric → Average of phonon momentum should be zero at equilibrium Thermal conductivity is infinite N process can be neglected in terms of thermal transport Phonon flux is the heat flow Net phonon flux is conserved throughout the system → Nothing impedes the flow of phonon momentum (no resistance)

Phonon-Phonon Case 2 : Umklapp Process involved Phonon Scattering Phonon-Phonon Case 2 : Umklapp Process involved U processes do not conserve net phonon momentum U processes are more frequent at higher temperatures U process must involve at least one phonon that has wavevector size comparable to the Brillouin zone At high temperature, high frequency modes are excited (BE distribution), resulting in more phonons available for U process k1 k2 G k3 k1 + k2 ky kx U processes resists the phonon momentum flux Scattering rate for U process determines the thermal conductivity (G.P. Srivastava, 1990) (G. Chen, 2005)

Phonon-Defect Phonon-Defect Interaction Scattering Rate Phonon Scattering Phonon-Defect Phonon-Defect Interaction Impurities, vacancies, dislocations Defects influence the mean free path of phonons by altering local acoustic impedance Elastic scattering Although magnitude of phonon wavevector does not change(elastic), the direction of the wave propagation changes As a consequence, net phonon momentum flux is not conserved Scattering Rate Independent of temperature Contribution to heat resistance is significant at low temperature Wavelength of phonons increases at low temperature, number of phonons with wavelength comparable to the defect radius increases Rayleigh law

Anharmonic Effects Thermal Conductivity vs. Temperature Phonon Scattering Anharmonic Effects Thermal Conductivity vs. Temperature A ~ B : κ ~ T 3 at low temperatures Most of the phonons have wavelength larger than the system size and defect size Temperature independent scattering processes(defect / boundary) are dominant → Phonon mean free path is constant Thermal conductivity is proportional to specific heat with T3 dependence B ~ D : U process significant Point where scattering rate of U process is frequent enough to yield phonon mean free path shorter than the size parameters Scattering rate of U process increase exponentially κ (log) T (log) A B C D C : Maximum point D ~ : κ ~ T −x at high temperatures Specific heat becomes constant (Dulong-Petit) Number of phonons available for U processes proportional to T 1 ~ T 2

Anharmonic Effects Thermal Conductivity vs. Temperature cv nU κ Phonon Scattering Anharmonic Effects Thermal Conductivity vs. Temperature κ T (log) A B C D cv nU

Phonon-Electron Phonon-Electron Scattering Phonon Scattering Phonon-Electron Phonon-Electron Scattering Lattice vibration distorts the electron wave function Phonon absorption/emission takes place Associated with Joule heating Energy transfer between electrons and phonons Momentum of electrons and phonons are crystal momentum Conservation of energy and crystal momentum Dominant scattering mechanism for electrons in metals Electron-electron scattering is negligible compared to electron-phonon scattering Electron-defect scattering is important at low temperatures

Phonon-Electron Scattering Rate Phonon Scattering Phonon-Electron Scattering Rate Electron-phonon scattering rate is inversely proportional to temperature at high temperatures At temperature higher than Debye temperature, number of phonons is proportional to temperature Number of electrons remain unchanged Electron energy > phonon energy Acoustic phonon has low energy compared to electron energy → can be neglected Significant at high temperature where optical phonons are excited Inelastic scattering (U process) Contribution of optical phonons are dominant

Phonon-Electron Transport Properties (Metals) Electrical resistance Phonon Scattering Phonon-Electron Transport Properties (Metals) Electrical resistance at low temperatures at high temperatures Electrical conductivity: proportional to T at high temperatures Drude-Lorentz expression Thermal conductivity: nearly constant at high temperatures Kinetic theory

Phonon-Photon-Electron Phonon Scattering Phonon-Photon-Electron Inter-band Transition (Indirect Semiconductors) Electron excitation by incident radiation (photon) Electron/photon/phonon interaction Phonon is absorbed/emitted to provide sufficient momentum change for electron band transition Conservation of energy and momentum k E Direct semiconductor Indirect semiconductor kphonon Eg Eg

Phonon-Photon Raman Effect Phonon Scattering Phonon-Photon Raman Effect Frequency shift between incident photon and scattered photon induced by phonon-photon scattering Spectroscopy : Position of Δωphoton for peak intensity depend upon temperature Stokes shift: phonon is absorbed from the photon Anti-stokes shift: phonon is emitted into the photon Stokes Anti-Stokes intermediate energy Scattered photon Incident photon Scattered photon Incident photon final energy initial energy Absorbed Phonon Emitted Phonon initial energy final energy

Conclusion Phonon Dispersion Phonon Scattering Relation between ω(energy) vs. k(momentum) Acoustic branch: significant contribution to thermal conductivity Optical branch: high frequency, slow vg , significant at high temperature Thermal properties can be calculated from dispersion relation Phonon Scattering Conservation of energy and crystal momentum Phonon-Phonon (U Process) : impedes phonon momentum flux Phonon-Defect : elastic, significant at low temperature Phonon-Electron (metals) : dominant at high temperature, optical branch Phonon-Photon : Raman scattering, Stokes shift

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Free electrons (Metals) Phonon Scattering Anharmonic Effects Phonon Scattering and Thermal Resistance Mechanism Frequency Temperature Low (T < θD) High (T > θD) Boundary (Size) ω0 T −3 T 0 Defects ω4 T 1 U Process ω1~2 T −3 exp(−αθD / T ) T 1~2 Free electrons (Metals) ω1 T −2 at low temperatures at high temperatures

Dispersion Relation Introduction Phonon Scattering Real Crystals